Recent advances in attitude control designs based on a 4-DOF Lagrangian dynamics have been reported in [1]}, [2]}. The Lagrangian dynamics is derivated by appending the holonomic constraints of a unit quaternion to the Lagrangian using Lagrange multipliers leading to Euler-Lagrange equations that include Lagrange multiplies, which are considered in conjunction with the algebraic constraint equations [3]}, [4]}, [5]}, [2]}. This approach obtains a set of configurations that satisfy the holonomic constraints embedded in \(\mathbb {R}^{4}\) . The standard variational methods is applied while the variations are constrained to respect the geometry of the configuration manifold. However, control designs based on energy-shaping methodology are not addressed in these works, since the energy-conservation property is not explored in this approach.
